wanminliu@gmail.com

Ext

open

boundary

•

¥6

-

Heroin

-

µlEy=x

The

btD

is

not

in

D.

y

=

✗

&

yzo

Y=

-

✗

&yzo--

A-

☐

:

I

;¥×]󲍻

y

so

g.

ix.

YI

has

meany

on

D

.

wanminliu@gmail.com

=¥

what's

ds

⑤

¥

,

#

ds

"

if

the

syne

is

given

by

a

graph

2-

=

2-

ix.

YI

,

where

4=1

ix.

y

,

HEIR

}

/

then

ds

=

i-zi-aytdxdy.x-yi-z-l.XZta.to/'#i+y'+z'--

1

.

and

a

is

the

distant

so

we

can

whet

from

the

observer

C-

the

center

✗

=

✗

142-1

,

4=4

7=2

of

the

moon

.

we

use

Y

and

2-

as

parameter

so

ds=*j+x

dydz

✗

=

-'

Sime

✗

is

positive

¥¥É÷

.

✗

y

=

I

(

1-

y

'

-

2-25

?

C-

4)

=

-

y

a-

yIzj±

✗

It

✗

z

=

÷

(

1-

y

?

2-

'

-22-1=-2-11

-

y

?zY¥

wanminliu@gmail.com

Tien

¥1

!ds=¥H÷÷_

pqgto-YF.net

dS=ÑÉx

dydz

we

give

a

description

of

=

.

dydt

tepwjetm

g-

Y

to

tie

7-

plane

Y

=

rcoso

2-

=

rsino

o

e

re

t

-

Eso

EI

=

dyda

ñ

I

=

¥µ

aydx

=

¥1

.

1.

±

.

=

¥

.

-4

"d

.

a

0

1-

F-

u

arms

fr om

tail

9-

→

=

f-

fat

.

d÷=&

WE

/

=

-441

-

E)

☐

wanminliu@gmail.com

why

dS=HZ×i-Ñdxdy

?

'

¥

-

let

the

surfer

by

Ñcx

-

y

)

=

IX.

Y

,

2-

ix.

Y

)

→p×

Then

Ñx

1×-77

=

11,0

,

2-

x

)

@

=

angle

between

r}

IX.

Y

)

=

10

,

I

,

2-7

)

rI

and

try

*

xril-lrillriis.mil

e-

=

ri.ru?--FxlYriill-wi01

a

-

5.

ri

=

Kii

-

lriiryiuioe-ri.rs

=*.ñlñ•ñ

-

lñ÷÷

=

T=

I

-

So

ds

=/

Fiery

/

=dxdy=Ea

dxdy

wanminliu@gmail.com

If

ÑCX

,

7)

=

I

✗

.

Y

,

2-

IX.

Y

)

then

ri

=

11

.

0

,

2-

x

)

try

=

10

,

I

,

Zy

)

so

=L

=

RI

-

RI

=

(

1,0

,

2×1

.

I

1.0

.

Zx

)

=

It

2-

I

G-

=

try

.

Try

=

I

0. 1 . 7

,

)

-110,1

,

241=11-2-5

,

,&

So

-Ñ=µxx) fttyJ-lZxZ

=ltai+zi-Ézaiy#󲍻

wanminliu@gmail.com

By

this

method

,

"

4-

we

pick

the

surface

F-

=

rj.ro?as x--XYY.Z)G--Ir.rI

go

the

svfneitShtnbY.E@Iz-1-xyi-xaTrTY.t_lXlY.Z1.Y

,

2-

)

try

=

I

Xy

,

I

,

0

)

Pz

=

I

✗

2-

,

0

,

I

7

wanminliu@gmail.com

Pet

R

be

the

region

bounded

by

the

①

compute

1St

xydv

or

by

using

the

cylinder

coordinate

cylinder

5+5=1

and

the

planes

2-

=

1

,

7=0

S¥.oylindercoodtnaÉ

in

the

font

octant

(

r

,

O

.

2-

)

coordinate

in

the

plane

f-

DV

=

ds

.

It

✗

=

room

_#

=

"

rdrdcidz-Y-rsho-xd-rdr-rfooi.no

)

󲍻÷÷i÷"¥

✗

<

a

wanminliu@gmail.com

=

if

&

-

I

③

compute

fssxydy

or

by

Gaius

divergence

Theorem

.

Gauss

divergence

theorem

.

f-

-4

!

"

)

¥

gggg

,

w=☒É*s

=

f-

E)

fast

.

󲍻

-

ciao

-

É

-

ix.

=

-1-41×1-21

✗

É

=

I

r

s

☐

wanminliu@gmail.com

S

=

S

,

U

Seuss

vsx

Uss

we

need

a

vertu

field

É

is

the

boon

day

not

R

.

go

that

dw

=p

=

XY

For

eah

Si

,

we

can

denote

its

unit

normal

veto

by

Ñ

,

we

can

pick

up

É=

OF

-105-1

xx

É

.

-

,

?

abaefdises-divf-C-x.g-y.bz

)

(0, 9×77)=117

.

So

-

in

#

-

plane

:*

:;¥ ¥ ± :¥ É÷É¥ ÷

.

am

Ifl

dived

V

=

☒

E.

Ñds

n

s

✗

5g

(

front

cylinder

)

S

,

-

bottom

Edina

wanminliu@gmail.com

Then

Ñ

,

=

(

0,0

,

-1

)

IffdivÉdv=ÉifÉ.Ñds

Ñ

,

=

(

o

,

o

,

1)

n

si

Ñ

,

=

I

-1

,

0,0

)

the

3rd

component

"

=

"

Ñ

NJ

=L

,

0

)

is

0

.

became

His

so

=P

.

Ñi

=

0

1=13,4

,

5

in

the

✗

y

-

plane

É

.

Ñz=

Xyz

/

(

2-

=

const

)

.

For

5

,

=

10.0.0

)

.

2-

=/

=D

xyds

=

If

✗

Yds

=)

!

f÷r

>

saoaodrdo

=

f-

q

✗

Ey

's

/

"

✗

→

%

armoire

,

y=rs W

osos

E

☐

(7--1)

wanminliu@gmail.com

¥-4k

be

the

intersection

of

thigh

theorist

Kroupa

.

and

origin

is

the

plane

9=7

and

the

sphere

center

of

the

sphere

✗

Yy

'

-12-1-1

Sette

intersects

in

the

director

counterclockwise

is

a

big

circle

.

with

respect

to

2-

-

axis

.

1×45+2-2=1

󲍻

1×7-22-41

①

compute

the

integral

7=2

Y=

2-

✗

=

WI

§

xyzdz

|

✗

=

v0

g

,

nanny

.my

,

gun

,

÷?g󲍻

y

,

÷

,

•

µ

,

1-

2-

=

¥

since

It

=

¥

cow

do

②

compute

the

same

integral

so

§

✗

Hdz

=/

o

"

v0

ftp.sina/Krsinojtrirod0

1-

by

stocks

's

theorem

.

=

z-f.fi

"

@

sojlsmojdo

wanminliu@gmail.com

22

=

¥

#

f.

Eggnog

.

do

¥met€

We

wan t

to

use

stocks

:

Theon

5h20

☒

we

pick

up

the

vertov

field

|

fgynzoj

=

¥40

É

=

(

o

,

o

,

XYZ

)

d

}

=L

dx.dy.dz

)

=

¥1

.

"

;

so

☒

•

DE

=

Xyz

dz

=

*

.

£

.

2k

=

a

§

.

di

=

If

☐

✗

E.

Ñds

1-

9

s

stocks

's

Theorem

wanminliu@gmail.com

So

Ñ=

10

.

¥

)

where

the

boundy

of

the

Sunfire

S

is

the

closet

yxÉ=anlÉ

ii.

I

curve

T

.

i

sufo

S

/

o

,

o

,

XYZ

s

"

*

=/

÷

:|

0*+14-12=0

=

2K¥

T

-

3×1×42-1

'T

a

normal

dtvatn

{

is

(

0

,

I

,

-

1)

=

IXZ

,

-77

,

0

)

the

plane

2=4

or

10

,

-1

,

a)

So

✗

E)

•

Ñ

we

need

the

3-

rdumpwtt

be

2=1× 7

,

-47,0110

,

-

¥

,

¥

)

polka

,

so

we

pick

up

Ñ=

=

7¥

4

unit

normal

vertu

feud

wanminliu@gmail.com

I

}

✗

E)

•

Ñds

=

If

as

ñ+y

:-c

7=2

S

The

plane

z=y

.

SIT

'd×dyµ÷w Y,¥

X'

+

visa

dS=HZ×4 Ñ=dxdy

=

,+ d×q= rÉ⇐""¥

"

am

§¥ds=ss÷→

.

,

÷÷÷÷÷÷

=

a

¥¥

☐